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Class 12 RD Sharma Solutions – Chapter 11 Differentiation – Exercise 11.2 | Set 1

  • Last Updated : 14 Jul, 2021

Question 1. Differentiate y = sin (3x + 5) with respect to x.

Solution:

We have,

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y = sin (3x + 5)



On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\sin\left( 3x + 5 \right)

On using chain rule, we have

\frac{d y}{d x} = \cos\left( 3x + 5 \right)\frac{d}{dx}\left( 3x + 5 \right)

\frac{d y}{d x} = \cos\left( 3x + 5 \right) \times 3

\frac{d y}{d x} = 3\cos\left( 3x + 5 \right)

Question 2. Differentiate y = tan2 x with respect to x.

Solution:

We have,



y = tan2 x

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}(\tan^2 x)

On using chain rule, we have

\frac{d y}{d x} = 2 \tan x\frac{d}{dx}\left( \tan x \right)

\frac{d y}{d x} = 2 \tan x \times \sec^2 x

\frac{d y}{d x} = 2 \tan x\sec^2 x

Question 3. Differentiate y = tan (x + 45°) with respect to x.

Solution:

We have,

y = tan (x + 45°)



y = \tan\left\{ \left( x + 45 \right)\frac{\pi}{180} \right\}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\tan\left\{ \left( x + 45 \right)\frac{\pi}{180} \right\}

On using chain rule, we have

\frac{d y}{d x} = \sec^2 \left\{ \left( x + 45 \right)\frac{\pi}{180} \right\} \times \frac{d}{dx}\left( x + 45 \right)\frac{\pi}{180}

\frac{d y}{d x} = \frac{\pi}{180} \sec^2 \left( x^\circ + 45^\circ \right)

Question 4. Differentiate y = sin (log x) with respect to x.

Solution:

We have,

y = sin (log x)

On differentiating y with respect to x we get,



\frac{d y}{d x} = \frac{d}{dx}\sin\left( \log x \right)

On using chain rule, we have

\frac{d y}{d x} = \cos\left( \log x \right)\frac{d}{dx}\left( \log x \right)

\frac{d y}{d x} = \frac{1}{x}\cos\left( \log x \right)

Question 5. Differentiate y = esin √x with respect to x.

Solution:

We have,

y = esin √x

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left( e^{\sin \sqrt{x}} \right)

On using chain rule, we have



\frac{d y}{d x} = e^{\sin \sqrt{x}} \frac{d}{dx}\left( \sin\sqrt{x} \right)

On using chain rule again, we have

\frac{d y}{d x} = e^{\sin \sqrt{x}} \times \cos\sqrt{x}\frac{d}{dx}\sqrt{x}

\frac{d y}{d x} = e^{\sin \sqrt{x}} \times \cos\sqrt{x} \times \frac{1}{2\sqrt{x}}

\frac{d y}{d x} = \frac{\cos\sqrt{x} e^{\sin}\sqrt{x}}{2\sqrt{x}}

Question 6. Differentiate y = etan x with respect to x.

Solution:

We have,

y = etan x

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left( e^{\tan x} \right)



On using chain rule, we have

\frac{d y}{d x} = e^{\tan x} \frac{d}{dx}\left( \tan x \right)

\frac{d y}{d x} = e^{\tan x}\sec^2 x

Question 7. Differentiate y = sin2 (2x + 1) with respect to x. 

Solution:

We have,

y = sin2 (2x + 1)

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left[ \sin^2 \left( 2x + 1 \right) \right]

On using chain rule, we have

\frac{d y}{d x} = 2\sin\left( 2x + 1 \right)\frac{d}{dx}\sin\left( 2x + 1 \right)



On using chain rule again, we have

\frac{d y}{d x} = 2\sin\left( 2x + 1 \right) \cos\left( 2x + 1 \right) \frac{d}{dx}\left( 2x + 1 \right)

\frac{d y}{d x} = 4\sin\left( 2x + 1 \right) \cos\left( 2x + 1 \right)

As sin 2A = 2 sin A cos A, we get

\frac{d y}{d x} = 2\sin2\left( 2x + 1 \right)

\frac{d y}{d x} = 2 \sin\left( 4x + 2 \right)

Question 8. Differentiate y = log7 (2x − 3) with respect to x.

Solution:

We have,

y = log7 (2x − 3)

As \log_a b = \frac{\log b}{\log a}, we have



y = \frac{\log\left( 2x - 3 \right)}{\log7}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{1}{\log7}\frac{d}{dx}\left\{ \log\left( 2x - 3 \right) \right\}

On using chain rule, we have

\frac{d y}{d x} = \frac{1}{\log7} \times \frac{1}{\left( 2x - 3 \right)}\frac{d}{dx}\left( 2x - 3 \right)

\frac{d y}{d x} = \frac{2}{\left( 2x - 3 \right)\log7}

Question 9. Differentiate y = tan 5x° with respect to x.

Solution:

We have,

y = tan 5x°

y = \tan\left( 5x \times \frac{\pi}{180} \right)

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\tan\left( 5x \times \frac{\pi}{180} \right)

On using chain rule, we have

\frac{d y}{d x} = \sec^2 \left( 5x \times \frac{\pi}{180} \right)\frac{d}{dx}\left( 5x \times \frac{\pi}{180} \right)

\frac{d y}{d x} = \left( \frac{5\pi}{180} \right) \sec^2 \left( 5x \times \frac{\pi}{180} \right)

\frac{d y}{d x} = \frac{5\pi}{180} \sec^2 \left( 5x^\circ\right)

Question 10. Differentiate y = 2^{x^3} with respect to x.

Solution:

We have,

y = 2^{x^3}

On differentiating y with respect to x we get,



\frac{d y}{d x} = \frac{d}{dx}\left( 2^{x^3} \right)

On using chain rule, we have

\frac{d y}{d x} = 2^{x^3} \times \log_e 2\frac{d}{dx}\left( x^3 \right)

\frac{d y}{d x} = 3 x^2 \times 2^{x^3} \times \log_e 2

Question 11. Differentiate y = 3^{e^x} with respect to x.

Solution:

We have,

y = 3^{e^x}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left( 3^{e^x} \right)

On using chain rule, we have

\frac{d y}{d x} = 3^{e^x} \log3\frac{d}{dx}\left( e^x \right)

\frac{d y}{d x} = e^x \times 3^{e^x} \log3

Question 12. Differentiate y = logx 3 with respect to x.

Solution:

We have,

y = logx 3

As \log_a b = \frac{\log b}{\log a}, we get

y = \frac{\log3}{\log x}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left( \frac{\log3}{\log x} \right)

\frac{d y}{d x} = \log3\frac{d}{dx} \left( \log x \right)^{- 1}



On using chain rule, we have

\frac{d y}{d x} = \log3 \times \left[ - 1 \left( \log x \right)^{- 2} \right]\frac{d}{dx}\left( \log x \right)

\frac{d y}{d x} = - \frac{\log3}{\left( \log x \right)^2} \times \frac{1}{x}

\frac{d y}{d x} = - \left( \frac{\log3}{\log x} \right)^2 \times \frac{1}{x} \times \frac{1}{\log3}

As \frac{\log b}{\log a} = \log_a b, we get

\frac{d y}{d x} = - \frac{1}{x\log3 \left( \log_3 x \right)^2}

Question 13. Differentiate y = 3^{x^2 + 2x} with respect to x.

Solution:

We have,

y = 3^{x^2 + 2x}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left( 3^{x^2 + 2x} \right)

On using chain rule, we have

\frac{d y}{d x} = 3^{x^2 + 2x} \times \log_e 3\frac{d}{dx}\left( x^2 + 2x \right)

\frac{d y}{d x} = \left( 2x + 2 \right) 3^{x^2 + 2x} \log_e 3

Question 14. Differentiate y = \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} with respect to x.

Solution:

We have,

y = \sqrt{\frac{a^2 - x^2}{a^2 + x^2}}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left( \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} \right)

On using chain rule, we have



\frac{d y}{d x} = \frac{1}{2} \left( \frac{a^2 - x^2}{a^2 + x^2} \right)^{\frac{1}{2} - 1} \times \frac{d}{dx}\left( \frac{a^2 - x^2}{a^2 + x^2} \right)

\frac{d y}{d x} = \frac{1}{2} \left( \frac{a^2 - x^2}{a^2 + x^2} \right)^\frac{- 1}{2} \times \left\{ \frac{\left( a^2 + x^2 \right)\frac{d}{dx}\left( a^2 - x^2 \right) - \left( a^2 - x^2 \right)\frac{d}{dx}\left( a^2 + x^2 \right)}{\left( a^2 + x^2 \right)^2} \right\}

\frac{d y}{d x} = \frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 2x\left( a^2 + x^2 \right) - 2x\left( a^2 - x^2 \right)}{\left( a^2 + x^2 \right)^2} \right\}

\frac{d y}{d x} = \frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 2x a^2 - 2 x^3 - 2x a^2 + 2 x^3}{\left( a^2 + x^2 \right)^2} \right\}

\frac{d y}{d x} = \frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 4x a^2}{\left( a^2 + x^2 \right)^2} \right\}

\frac{d y}{d x} = \frac{- 2x a^2}{\sqrt{a^2 - x^2} \left( a^2 + x^2 \right)^\frac{3}{2}}

Question 15. Differentiate y = 3^{x \log x} with respect to x.

Solution:

We have,

y = 3^{x \log x}

On differentiating y with respect to x we get,



\frac{d y}{d x} = \frac{d}{dx}\left( 3^{x \log x} \right)

On using chain rule, we have

\frac{d y}{d x} = 3^{x \log x} \times \log_e 3\frac{d}{dx}\left( x \log x \right)

\frac{d y}{d x} = 3^x \log x \times \log_e 3\left[ x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx}\left( x \right) \right]

\frac{d y}{d x} = 3^{x \log x} \times \log_e 3\left[ \frac{x}{x} + \log x \right]

\frac{d y}{d x} = 3^{x \log x} \left( 1 + \log x \right) \times \log_e 3

\frac{d y}{d x} = 3^{x \log x} \left( 1 + \log x \right)\log_e 3

Question 16. Differentiate y = \sqrt{\frac{1 + \sin x}{1 - \sin x}}    with respect to x.

Solution:

We have,

y = \sqrt{\frac{1 + \sin x}{1 - \sin x}}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx} \left( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \right)

\frac{d y}{d x} = \frac{d}{dx} \left( \frac{1 + \sin x}{1 - \sin x} \right)^\frac{1}{2}

On using chain rule, we have

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 + \sin x}{1 - \sin x} \right)^{\frac{1}{2} - 1} \frac{d}{dx}\left( \frac{1 + \sin x}{1 - \sin x} \right)

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 - \sin x}{1 + \sin x} \right)^\frac{1}{2} \left[ \frac{\left( 1 - \sin x \right)\left( \cos x \right) - \left( 1 + \sin x \right)\left( - \cos x \right)}{\left( 1 - \sin x \right)^2} \right]

\frac{d y}{d x} = \frac{1}{2}\frac{\left( 1 - \sin x \right)^\frac{1}{2}}{\left( 1 + \sin x \right)^\frac{1}{2}}\left[ \frac{\cos x - \cos x \sin x + \cos x + \sin x \cos x}{\left( 1 - \sin x \right)^2} \right]

\frac{d y}{d x} = \frac{1}{2} \times \frac{2\cos x}{\sqrt{1 + \sin x}\left( 1 - \sin x \right)\frac{3}{2}}

\frac{d y}{d x} = \frac{\cos x}{\sqrt{1 + \sin x}\left( 1 - \sin x \right)\frac{3}{2}}

\frac{d y}{d x} = \frac{\cos x}{\sqrt{1 + \sin x}\sqrt{1 - \sin x}\left( 1 - \sin x \right)}



\frac{d y}{d x} = \frac{\cos x}{\sqrt{1 - \sin^2 x} \times \left( 1 - \sin x \right)}

\frac{d y}{d x} = \frac{\cos x}{\cos x\left( 1 - \sin x \right)}

\frac{d y}{d x} = \frac{1}{\left( 1 - \sin x \right)} \times \frac{\left( 1 + \sin x \right)}{\left( 1 + \sin x \right)}

\frac{d y}{d x} = \frac{\left( 1 + \sin x \right)}{\left( 1 - \sin^2 x \right)}

\frac{d y}{d x} = \frac{1 + \sin x}{\cos^2 x}

\frac{d y}{d x} = \frac{1}{\cos x}\left( \frac{1}{\cos x} + \frac{\sin x}{\cos x} \right)

\frac{d y}{d x} = \sec x\left( \sec x + \tan x \right)

Question 17. Differentiate y = \sqrt{\frac{1 - x^2}{1 + x^2}}    with respect to x.

Solution:

We have,

y = \sqrt{\frac{1 - x^2}{1 + x^2}}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx} \left( \sqrt{\frac{1 - x^2}{1 + x^2}} \right)

\frac{d y}{d x} = \frac{d}{dx} \left( \frac{1 - x^2}{1 + x^2} \right)^\frac{1}{2}

On using chain rule, we have

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 - x^2}{1 + x^2} \right)^{\frac{1}{2} - 1} \times \frac{d}{dx}\left( \frac{1 - x^2}{1 + x^2} \right)

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 - x^2}{1 + x^2} \right)^\frac{- 1}{2} \times \left\{ \frac{\left( 1 + x^2 \right)\frac{d}{dx}\left( 1 - x^2 \right) - \left( 1 - x^2 \right)\frac{d}{dx}\left( 1 + x^2 \right)}{\left( 1 + x^2 \right)^2} \right\}

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 + x^2}{1 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 2x\left( 1 + x^2 \right) - 2x\left( 1 - x^2 \right)}{\left( 1 + x^2 \right)^2} \right\}

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 + x^2}{1 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 2x - 2 x^3 - 2x + 2 x^3}{\left( 1 + x^2 \right)^2} \right\}

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 + x^2}{1 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 4x}{\left( 1 + x^2 \right)^2} \right\}

\frac{d y}{d x} = \frac{- 2x}{\sqrt{1 - x^2} \left( 1 + x^2 \right)^\frac{3}{2}}



Question 18. Differentiate y = (log sin x)2 with respect to x.

Solution:

We have,

y = (log sin x)2

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx} \left( \log \sin x \right)^2

On using chain rule, we have

\frac{d y}{d x} = 2\left( \log \sin x \right)\frac{d}{dx}\left( \log \sin x \right)

\frac{d y}{d x} = 2\left( \log \sin x \right) \times \frac{1}{\sin x}\frac{d}{dx}\left( \sin x \right)

\frac{d y}{d x} = 2\left( \log \sin x \right) \times \frac{1}{\sin x} \times \cos x

\frac{d y}{d x} = 2\left( \log \sin x \right)\cot x

Question 19. Differentiate y = \sqrt{\frac{1 + x}{1 - x}}    with respect to x.

Solution:

We have,

y = \sqrt{\frac{1 + x}{1 - x}}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx} \left( \sqrt{\frac{1 + x}{1 - x}} \right)

\frac{d y}{d x} = \frac{d}{dx} \left( \frac{1 + x}{1 - x} \right)^\frac{1}{2}

On using chain rule, we have

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 + x}{1 - x} \right)^{\frac{1}{2} - 1} \times \frac{d}{dx}\left( \frac{1 + x}{1 - x} \right)

On using quotient rule, we have

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 + x}{1 - x} \right)^\frac{- 1}{2} \times \left\{ \frac{\left( 1 - x \right)\frac{d}{dx}\left( 1 + x \right) - \left( 1 + x \right)\frac{d}{dx}\left( 1 - x \right)}{\left( 1 - x \right)^2} \right\}



\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 - x}{1 + x} \right)^\frac{1}{2} \left\{ \frac{\left( 1 - x \right)\left( 1 \right) - \left( 1 + x \right)\left( - 1 \right)}{\left( 1 - x \right)^2} \right\}

\frac{d y}{d x} = \frac{1}{2} \left( \frac{1 - x}{1 + x} \right)^\frac{1}{2} \left\{ \frac{1 - x + 1 + x}{\left( 1 - x \right)^2} \right\}

\frac{d y}{d x} = \frac{1}{2}\frac{\left( 1 - x \right)^\frac{1}{2}}{\left( 1 + x \right)^\frac{1}{2}} \times \frac{2}{\left( 1 - x \right)^2}

\frac{d y}{d x} = \frac{1}{\sqrt{1 + x} \left( 1 - x \right)^\frac{3}{2}}

Question 20. Differentiate y = \sin \left( \frac{1 + x^2}{1 - x^2} \right)    with respect to x.

Solution:

We have,

y = \sin \left( \frac{1 + x^2}{1 - x^2} \right)

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx} \left( \sin \left( \frac{1 + x^2}{1 - x^2} \right) \right)

On using chain rule, we have

\frac{d y}{d x} = \cos x\left( \frac{1 + x^2}{1 - x^2} \right)\frac{d}{dx}\left( \frac{1 + x^2}{1 - x^2} \right)

On using quotient rule, we have

\frac{d y}{d x} = \cos x\left( \frac{1 + x^2}{1 - x^2} \right)\left[ \frac{\left( 1 - x^2 \right)\frac{d}{dx}\left( 1 + x^2 \right) - \left( 1 + x^2 \right)\frac{d}{dx}\left( 1 - x^2 \right)}{\left( 1 - x^2 \right)^2} \right]

\frac{d y}{d x} = \cos x\left( \frac{1 + x^2}{1 - x^2} \right)\left[ \frac{\left( 1 - x^2 \right)\left( 2x \right) - \left( 1 + x^2 \right)\left( - 2x \right)}{\left( 1 - x^2 \right)^2} \right]

\frac{d y}{d x} = \cos x\left( \frac{1 + x^2}{1 - x^2} \right)\left[ \frac{2x - 2 x^3 + 2x + 2 x^3}{\left( 1 - x^2 \right)^2} \right]

\frac{d y}{d x} = \frac{4x}{\left( 1 - x^2 \right)^2}\cos x\left( \frac{1 + x^2}{1 - x^2} \right)

Question 21. Differentiate y = e^{3x} \cos2x with respect to x.

Solution:

We have,

y = e^{3x} \cos2x

On differentiating y with respect to x we get,



\frac{d y}{d x} = \frac{d}{dx} \left( e^{3x} \cos2x\right)

On using product rule, we have

\frac{d y}{d x} = e^{3x} \times \frac{d}{dx}\left( \cos2x \right) + \cos2x\frac{d}{dx}\left( e^{3x} \right)

On using chain rule, we have

\frac{d y}{d x} = e^{3x} \times \left( - \sin2x \right)\frac{d}{dx}\left( 2x \right) + \cos2x e^{3x} \frac{d}{dx}\left( 3x \right)

\frac{d y}{d x} = - 2 e^{3x} \sin2x + 3 e^{3x} \cos2x

\frac{d y}{d x} = e^{3x} \left( 3 \cos2x - 2 \sin2x \right)

Question 22. Differentiate y = sin(log sin x) with respect to x.

Solution:

We have,

y = sin(log sin x)

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx} \left(sin(log sin x)\right)

On using chain rule, we have

\frac{d y}{d x}=\cos(\log \sin x)\frac{d}{dx}(\log \sin x)

On using chain rule again, we have

\frac{d y}{d x}=\cos (\log \sin x)\frac{1}{sin x}\frac{d}{dx}(\sin x)

\frac{d y}{d x}=\cos (\log \sin x)\frac{cos x}{sin x}

\frac{d y}{d x}=\cos (\log \sin x) \cot x

Question 23. Differentiate y = etan 3x with respect to x.

Solution:

We have,

y = etan 3x

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left(e^{\tan3 x} \right)

On using chain rule, we have

\frac{d y}{d x} = e^{\tan3x} \frac{d}{dx}\left( \tan3x \right)

\frac{d y}{d x} = e^{\tan3x} \sec^2 3x \times \frac{d}{dx}\left( 3x \right)

\frac{d y}{d x} = e^{\tan3x} \sec^2 3x \times 3

\frac{d y}{d x} = 3e^{\tan3x} \sec^2 3x

Question 24. Differentiate y = e^{\sqrt{\cot x}}    with respect to x.

Solution:

We have,



y = e^{\sqrt{\cot x}}

On differentiating y with respect to x we get,

\frac{d y}{d x} = \frac{d}{dx}\left(e^{\sqrt{\cot x}} \right)

\frac{d y}{d x} = \frac{d}{dx}\left( e^{\left( \cot x \right)^\frac{1}{2} }\right)

On using chain rule, we have

\frac{d y}{d x} = e^{\left( \cot x \right)^\frac{1}{2}} \times \frac{d}{dx} \left( \cot x \right)^\frac{1}{2}

\frac{d y}{d x} = e^{\sqrt{\cot x}} \times \frac{1}{2} \left( \cot x \right)^{\frac{1}{2} - 1} \frac{d}{dx}\left( \cot x \right)

\frac{d y}{d x} = - \frac{e^{\sqrt{\cot x}}{cosec}^2 x}{2\sqrt{\cot x}}




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